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Properties

Label 2-546-273.242-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.660 - 0.750i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−1.71 + 0.224i)3-s + (0.866 + 0.499i)4-s + (−2.30 − 0.616i)5-s + (−1.71 − 0.227i)6-s + (2.49 + 0.885i)7-s + (0.707 + 0.707i)8-s + (2.89 − 0.770i)9-s + (−2.06 − 1.19i)10-s + (−4.42 + 1.18i)11-s + (−1.59 − 0.664i)12-s + (−1.86 + 3.08i)13-s + (2.17 + 1.50i)14-s + (4.09 + 0.542i)15-s + (0.500 + 0.866i)16-s + (−2.95 + 5.11i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.991 + 0.129i)3-s + (0.433 + 0.249i)4-s + (−1.02 − 0.275i)5-s + (−0.700 − 0.0930i)6-s + (0.942 + 0.334i)7-s + (0.249 + 0.249i)8-s + (0.966 − 0.256i)9-s + (−0.652 − 0.376i)10-s + (−1.33 + 0.357i)11-s + (−0.461 − 0.191i)12-s + (−0.516 + 0.856i)13-s + (0.582 + 0.401i)14-s + (1.05 + 0.140i)15-s + (0.125 + 0.216i)16-s + (−0.716 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.660 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359659 + 0.795351i\)
\(L(\frac12)\) \(\approx\) \(0.359659 + 0.795351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (1.71 - 0.224i)T \)
7 \( 1 + (-2.49 - 0.885i)T \)
13 \( 1 + (1.86 - 3.08i)T \)
good5 \( 1 + (2.30 + 0.616i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.42 - 1.18i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.95 - 5.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.323 + 1.20i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.26 - 5.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.63iT - 29T^{2} \)
31 \( 1 + (5.95 - 1.59i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.75 + 0.468i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.95 - 1.95i)T - 41iT^{2} \)
43 \( 1 - 9.07iT - 43T^{2} \)
47 \( 1 + (0.736 - 2.74i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.46 + 1.99i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.8 + 3.18i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.99 + 5.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.93 + 2.66i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.41 + 8.41i)T - 71iT^{2} \)
73 \( 1 + (2.85 + 10.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.81 - 6.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.74 + 9.74i)T - 83iT^{2} \)
89 \( 1 + (2.47 - 9.24i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.90 - 3.90i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.24261553142424273944774446341, −10.74290438382642708263307581908, −9.414478434655179002514540810529, −8.085092231465126417903892606089, −7.52349730864985848806861267263, −6.46840895759225758159463332336, −5.21951792841943868761997066175, −4.77235496911525906203311167661, −3.80645451104045067327476394251, −1.94471480343520801262025287231, 0.44186456768838431661085673538, 2.48912287600273914568534198395, 3.90833444470383014241363933233, 5.02587031945848271308691022587, 5.38912105304819652314582531502, 7.00838972571116074039712325281, 7.44412180865944468604093299038, 8.420348220772669898456208540813, 10.14797637042703893334697260616, 10.86435948993217729663294454565

Graph of the $Z$-function along the critical line